The Chicago Board Option Exchange (CBOE) VIX index is a calculated measure of expected annualized 30-day variance as a measure of volatility for the S\&P 500. The VIX is expressed as the following:

\begin{equation}
VIX = 100 \sqrt{var}
\end{equation}

Where:
\begin{equation}
var = (365/30) \times \mbox{Expected 30-day variance}
\end{equation}
Expected 30-day variance is the sum of the squared standard deviations of the S\&P 500 rate of return during the 30 days:
\begin{equation}
\mbox{30-day variance} = {\sum{\sigma^2_{t}}}
\end{equation}

The S\&P500 index contains multiple companies that vary in market capitalizations, industries, book-to-market ratios and several other factors. The wide diversity in the S\&P500 generates variable effects on the index's overall performance. Each Fama-French portfolio utilized in this paper consists of a specialized basket of stocks with similar book-to-market ratios and sizes. In order to understand the relative impact the S\&P500's volatility has on each Fama-French portfolio, six linear regression models were constructed to measure the effect that the change in VIX had on each Fama-French portfolio's log return. 

However, the prediction using VIX models performs very poorly, as shown in Figure. \ref{fig:vix_expand_prec}. This is attributable to poor correlation between the change in VIX and Fama-French portfolio returns. Attempting to explain and predict specialized Fama-French portfolio returns using only average 30-day variance of the high variability population of the S\&P500 in a single factor linear regression model does not include enough explanatory data to improve model predictions. The average precision is only about 10 percent.

\begin{figure}[hbtp]
\centering
\includegraphics[width=7.5cm, height=5cm]{../results/vix_expand_precision.png}
\caption{The Overall Precision of Prediction of VIX Models using Expanding Window}
\label{fig:vix_expand_prec}
\end{figure}